Brooks-type colourings of digraphs in linear time

2024-05-08

arXiv.org

AI Summary

This paper addresses the generalization of Brooks’ theorem to directed graphs, resolving the absence of linear-time constructive algorithms for existing generalized coloring models—such as variable weak degeneracy. We introduce two novel concepts: *variable bidegeneracy* and *F-dicolouring*, enabling a unified Brooks-type theorem that characterizes list coloring, variable degenerate subgraph partitioning, and other generalized coloring frameworks on digraphs. Crucially, we design the first *O(n + m)*-time algorithm that simultaneously decides feasibility and constructs such colorings, ensuring both theoretical completeness and practical efficiency. Our framework extends the theoretical boundaries of digraph coloring theory and yields the first linear-time coloring scheme supporting diverse generalized constraints. This advances the algorithmic paradigm for structural digraph problems by unifying expressive modeling power with optimal computational efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Brooks' Theorem is a fundamental result on graph colouring, stating that the chromatic number of a graph is almost always upper bounded by its maximal degree. Lov'asz showed that such a colouring may then be computed in linear time when it exists. Many analogues are known for variants of (di)graph colouring, notably for list-colouring and partitions into subgraphs with prescribed degeneracy. One of the most general results of this kind is due to Borodin, Kostochka, and Toft, when asking for classes of colours to satisfy"variable degeneracy"constraints. An extension of this result to digraphs has recently been proposed by Bang-Jensen, Schweser, and Stiebitz, by considering colourings as partitions into"variable weakly degenerate"subdigraphs. Unlike earlier variants, there exists no linear-time algorithm to produce colourings for these generalisations. We introduce the notion of (variable) bidegeneracy for digraphs, capturing multiple (di)graph degeneracy variants. We define the corresponding concept of $F$-dicolouring, where $F = (f_1,...,f_s)$ is a vector of functions, and an $F$-dicolouring requires vertices coloured $i$ to induce a"strictly-$f_i$-bidegenerate"subdigraph. We prove an analogue of Brooks' theorem for $F$-dicolouring, generalising the result of Bang-Jensen et al., and earlier analogues in turn. Our new approach provides a linear-time algorithm that, given a digraph $D$, either produces an $F$-dicolouring of $D$, or correctly certifies that none exist. This yields the first linear-time algorithms to compute (di)colourings corresponding to the aforementioned generalisations of Brooks' theorem. In turn, it gives an unified framework to compute such colourings for various intermediate generalisations of Brooks' theorem such as list-(di)colouring and partitioning into (variable) degenerate sub(di)graphs.

Problem

• Extends Brooks' theorem to digraphs with variable bidegeneracy constraints
• Provides linear-time algorithm for F-dicolouring or certification of impossibility
• Unifies framework for computing degenerate subdigraph partitions and list-dicolourings

Innovation

• Introduces variable bidegeneracy for digraphs
• Provides linear-time F-dicolouring algorithm
• Unifies framework for multiple degeneracy variants